Mistakes give rise to Problems- 16

You can remove the brackets in a fraction (note that this is fraction, not the Legendre symbol) , just like in the following fraction \[\displaystyle \biggl(\color{Blue}{\dfrac{a}{b}} \biggr) = \color{Blue}{\dfrac{a}{b}}\]

But if you do that in the \(\color{Purple}{\textbf{Binomial Coefficient}}\), like \[\displaystyle \dbinom{a}{b} = \dfrac{a}{b}\] then it's a \(\color{Red}{\textbf{Big Mistake !!!}}\)

But for how many ordered pairs \(\color{Green}{(a,b)}\) such that \(0\leq a \leq 20\) and \(0\leq b \leq 20\) is the above said "false" property seen to be "true" ?

Details and assumptions

\(\bullet\quad\) Factorial (\(a!\)) is defined only for non-negative integers.

\(\bullet\quad\)\(\dbinom{a}{b} = \dfrac{a!}{b! \times (a-b)!}\)

\(\bullet\quad\)\(\dbinom{a}{b} = 0\) if \(a<b\).

\(\bullet\quad\)\(0! =1\)

This problem is a part of the set Mistakes give rise to Problems. Try other problems of the set too.


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